PoS(Texas 2010)040

3D Collapse and Accretion in Slowly RotatingPolytropes

Aldo Batta and William H. LeeInstituto de Astronomía, UNAM - Ciudad Universitaria, MéxicoE-mail: abatta@astro.unam.mx

Several tests have been made using the code GADGET-2, to model the collapse and accretionof a Polytropic envelope onto a BH of 1M. Based on previous results, a minimal resolution(NSPH = 50000) is chosen and simulations are carried out with different initial angular momen-tum distributions. These will provide information about the formation of accretion disks at dif-ferent distances from the BH. By observing the accretion of a non rotating, low internal energyUint polytropic envelope, we tested the Paczynski-Wiita potential used to account for some rel-ativistic effects on the gas due to the BH. These series of tests will help develop the necessarymodifications to GADGET-2 in the context of the collapsar model for GRBs.

25th Texas Symposium on Relativistic Astrophysics - TEXAS 2010December 06-10, 2010Heidelberg, Germany

c© Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/

PoS(Texas 2010)040

3D Collapse and Accretion in Slowly Rotating Polytropes

1. Introduction

Though collapse and accretion of rotating material in astrophysical systems is a quite commonissue, it hasn’t been yet completely studied. This is due to the complexity of the systems involved,which sometimes include a complicated EOS for the collapsing and accreted material, as well asnon trivial heating and cooling mechanisms that take place at high densities and temperatures. Dueto these physical complexities the differential equations for hydrodynamics are difficult to solveand this translates in only few works done in 3D which include rotation effects, self-gravity and theinstabilities that might come from them.

In order to make a “full” 3D study of the collapse and accretion for rotating material in anastrophysical problem, we consider the collapsar scenario [4] [3], which consists on a PreSN starwhose Fe core and outer envelope, are about to collapse due to the shutdown of nuclear reactionsin the central region and the Fe core reaching the Chandrasekhar mass. In our scenario, the Fecore collapses directly to form a BH which accretes the infalling outer envelope. To model gasdynamics, the simulations of this scenario will be made using the code GADGET-2 based on SPH(Smoothed Particle Hydrodynamics) [1]. In this work we show the current modifications to the codeand some tests made to account for the accretion onto a BH and the pseudo-relativistic potentialconsidered for the compact object.

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Figure 1: On the top panel the polytropic star column density after 21 dynamical time scales along the Y(left) and the Z (right) axis both without visible symmetry breaking. On the bottom the total energy (left)and angular momentum (right) with a very good conservation.

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3D Collapse and Accretion in Slowly Rotating Polytropes

2. Stability tests

Before starting with the collapsar model, we tested whether the code could properly solvethe stationary solution of a non-rotating γ = 5/3 polytrope, without having instabilities break theexpected spherical symmetry. For 2M polytropic stars with central density ρc ' 2.5×109 g cm−3

we tested over several dynamical time scales (at least 10) whether there was any visible symmetrybreaking or not. For these tests we used natural units by considering G = 1 and scaling all unitswith the mass Mp = 2M = 3.98× 1033 [g], length Rp = 1.3099× 108 [cm] and dynamical timescale tdyn = (R3

p/GMp)1/2 = 0.092 [s] of the system. Figure 1 shows the column density along the

Y (left) and Z (right) axis of such a polytropic star with γ = 5/3 after 21 dynamical time scales.The results show no visible symmetry breaking and there is a very good energy and total angularmomentum conservation of 1/100 and 1/100000 respectively (bottom panel of Figure 1).

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Figure 2: Normalized potential energy Epot (up-right), internal energy Eint (up-left), kinetic energy Ecin

(down-left) and total energy Etot (down-right)for the polytropic stars with different rotation rates. Red,black, blue and purple lines correspond to angular velocities Ω0 = 0.026,0.052,0.104,0.260 respectively.

In order to see if rotation could be of importance to break the spherical symmetry of this test,we repeated the calculation using different amounts of initial angular momentum (assuming rigidbody rotation around the Z axis). Figure 2 shows the internal, potential, kinetic and total energy ofsuch tests which indicates that increasing the initial angular momentum implies bigger oscillations(amplitudes) and a shift on the time the first minimum occurs in the internal energy. Figure 3 showsthe total angular momentum and the column density along the Y axis after 30 dinamical time scales.Overall, the results were qualitatively similar to the tests with no rotation, showing that the codecould properly evolve such configurations without generating huge instabilities. All these tests were

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PoS(Texas 2010)040

3D Collapse and Accretion in Slowly Rotating Polytropes

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Figure 3: Total angular momentum for tests with different initial rotation rates (left) and column densityalong the Y axis for the second slowest polytropic star (Ω = 0.52166) after 30 dynamical time scales (right).Rotation creates a small deviation from spherical symmetry.

made using 5×104 SPH particles given that previous results showed this resolution properly solvedthe shock after collapse and tests with 10 times this resolution showed no significant changes.

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Figure 4: Potential (up-right), internal Eint (up-left), kinetic (down-left) and total (down-right) energy plotsfor the collapse of γ = 5/3 polytropes with low internal energy (half the value needed for a stable solutionEint = Est/2). Each curve represents different rotation rates (rigid body angular velocity); purple line Ω =

2.0866, gray line Ω = 1.5650, black line Ω = 1.0433, blue line Ω = 0.52166 and red line Ω = 0.13041.

Working towards a collapsar model we proceeded to study the collapse of such polytropic stars

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3D Collapse and Accretion in Slowly Rotating Polytropes

with different rotation rates, by removing half the internal energy Est needed for a stable solution(i.e. decreasing the pressure). In Figure 4 we show collapses for different rotation rates (assuming arigid body angular momentum distribution) and how increasing angular momentum translates intoa shallower collapse shown on the potential and internal energies (figure 4 top panels).

3. Collapse and Accretion in Slowly Rotating Polytropes

In the context of the collapsar model, we tested the collapse and accretion of rotating polytropicstars considering that the innermost 1M formed a BH, with the outer layers collapsing due to thelack of pressure support. To account for accretion, we included a sink particle in the code, withan accretion radius racc = ηGMBH/c2, where η > 2 determines how far from the Schwarzschildradius accretion takes place and the gas particles are swallowed by the BH. Here we considered avalue of η = 10.

Figure 5: Density on the XZ plane for polytropes with different rotation rates after t = 2tdyn. The for-mation (and size) of the accretion disk depends on the initial angular momentum distribution given byΩ0 = 0.104331,0.93898 and 1.8258 from left to right.

Ω0 [Nat. units] J0 in CGS rc in CGS rc [Nat. units] racc [Nat. units]0.1043 5.27×1015 [cm2 s−1] 2.09×105 [ cm ] 0.001 0.010.9389 4.58×1016 [cm2 s−1] 1.58×107 [ cm ] 0.12 0.011.8258 9.24×1016 [cm2 s−1] 6.31×107 [ cm ] 0.48 0.01

Table 1: Initial specific angular momentum J0 for the innermost material and its corresponding circular-ization radius rc. The model with rC < racc (and also J < Jmin) shows spherical accretion and no disk isappreciated.

As we can see from Figure 5, if rotation is too slow the infalling material cannot form an accre-tion disk around the BH, and we end up with Bondi like accretion. On the other hand, increasingrotation results in the formation of accretion disks of different sizes and densities. Consideringthat the gas will eventually get into a circular keplerian motion around the BH at a distance rC, we

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PoS(Texas 2010)040

3D Collapse and Accretion in Slowly Rotating Polytropes

can estimate this circularization radius rC in terms of the initial specific angular momentum of theinnermost material:

rC ' J20/GMBH (3.1)

On table 1 we show an estimate of the circularization radius for polytropes with differentrotation rates shown on Figure 5. Notice how that the slowest polytrope has rC < racc, this preventsthe gas from forming a keplerian disk around the BH before being accreted. There is anothercondition for disk formation around a BH given by the innermost stable circular orbit at Risco = 3Rg.To stay in this orbit the gas must have a minimum angular momentum Jmin = 6GMBH/c. For a BHwith MBH = 1M the minimum angular momentum is Jmin = 2.6575×1016cm2s−1. So far we haveonly considered a Newtonian potential for the BH but to account for the existence of this innermoststable circular orbit we must consider a pseudo relativistic potential for the BH.

4. Modifications to the BH Potential

In order to account for some of the relativistic effects on the gas, we included a Paczynski-Wiita (PW) potential for the BH. We tested the PW potential by observing the collapse and accre-tion of a 1M non-rotating, polytropic envelope with very low internal energy (to avoid dispersionon the accelerations due to pressure gradients) and taking into account its self-gravity to compareits acceleration due to the 1M BH potential.

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Figure 6: Acceleration of gas particles with lowinternal energy, falling into a 1M BH with aPaczynski-Wiita potential. The blue solid line andthe black solid line represent the Newtonian and thePW accelerations (respectively) of a test particle,due to the BH. The red dots represent the accelera-tion of gas particles (considering self-gravity), andthe cyan dots represent the acceleration of gas par-ticles due only to the BH (without self-gravity).

As seen in Figure 6, the gas acceleration due to the BH (cyan dots) agrees with the accelerationexpected due to a 1M BH with a PW potential (black solid line). Now that the PW potential hasbecome fully functional, the gas would be able to reproduce the position of the innermost stablecircular orbit at a distance Risco = 6GMBH/c2 from the BH. So if we want to include this effecton the gas motion we have to consider an accretion radius racc equal or smaller than Risco. Fromtable 1, and Figure 5, we see that both the polytropes that formed a disk around the BH, have initialspecific angular momentum J0 bigger than the minimum angular momentum Jmin needed to stayon the innermost stable circular orbit. On future calculations we must include information aboutthe circularization radius rC observed on the simulations, the expected rC from the initial angulalmomenutm J0, and a comparison to the breakup angular momentum needed for the star to disruptby its own rotation. This information will lead us to a more realistic collapsar model.

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3D Collapse and Accretion in Slowly Rotating Polytropes

5. Summary

These tests have shown that the code GADGET-2 [1] is able to solve properly the collapse andaccretion of a polytropic envelope into a BH, and have taken us closer in constructing the collapsarmodel. Results show that given enough angular momentum, the gas would create an accretiondisk near the BH but, in order to be accreted, the material from the disk will have to lose angularmomentum, and this depends directly on how efficient the angular momentum transport from theinner, to the outer parts of the disk (viscosity) is. At the time, we have not considered any otherviscosity that the one coming from the SPH formulation numerically, but this would definitely bean issue to explore in future modifications. Further work will focus on including some simplecooling of the gas in order to observe its importance on determining the disk properties and thedevelopment of instabilities.

Acknowledgments

Support from CONACyT (83254 and a graduate fellowship for AB), the TEXAS Symposium2010 LOC, and assistance from A. Rodriguez-Gonzalez with Gadget-2 is gratefully acknowledged,as is the use of SPLASH v.1.13 by D.J.Price.

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